Solving Proportions by Cross Multiplication · PDF file[Operations and Computation Goal 2] ... per partnership: ... Algebraic Thinking Go over the answers to Problems 1–10. Review

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Lesson 8�3 703

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 64 – 68, 295 – 297

Key Concepts and Skills• Apply multiplication and division facts

to find cross products. [Operations and Computation Goal 2]

• Multiply whole or decimal numbers. [Operations and Computation Goal 2]

• Use cross products to write open number sentences. [Operations and Computation Goal 6]

• Describe rules for patterns and use them to solve problems. [Patterns, Functions, and Algebra Goal 1]

• Use a method to solve equations. [Patterns, Functions, and Algebra Goal 2]

Key ActivitiesStudents use the cross-products rule to determine whether two fractions are equivalent. They solve rate problems by writing proportions and using cross multiplication.

Ongoing Assessment: Informing Instruction See page 705.

Ongoing Assessment: Recognizing Student Achievement Use journal page 288. [Operations and Computation Goal 6]

Key Vocabularycross products � cross multiplication

MaterialsMath Journal 2, pp. 286, 288–289BStudy Link 8�2calculator (optional)

Playing Fraction/Whole Number Top-ItStudent Reference Book, pp. 319 and 320per partnership: 4 each of number cards 1–10 (from the Everything Math Deck, if available), calculator (optional)Students practice calculating and comparing products of fractions and whole numbers.

Math Boxes 8�3Math Journal 2, p. 287 Students practice and maintain skillsthrough Math Box problems.

Study Link 8�3Math Masters, pp. 249 and 251 Students practice and maintain skillsthrough Study Link activities.

READINESS

Solving Equations (ax = b)Students practice solving simple equations.

ENRICHMENT Using Double Number LinesMath Masters, pp. 249A and 249BStudents use double number lines to solve rate problems.

EXTRA PRACTICE Calculating Ingredient AmountsMath Masters, p. 250Students practice solving rate problems by calculating ingredient amounts for a recipe.

ELL SUPPORT Illustrating Termsposterboard � markersStudents make posters illustrating how to use cross products to solve open proportions.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Solving Proportionsby Cross Multiplication

Objective To introduce and use cross multiplication to solve proportions.t

��

Common Core State Standards

703_EMCS_T_TLG2_G6_U08_L03_576922.indd 703 2/22/11 10:43 AM

Date Time

Math Message

For Part a of each problem, write = or ≠ in the answer box. For Part b, calculate the cross products.

1. a. 3 _ 5 = 6 _ 10 2. a. 7 _ 8 2 _ 3

b. b.

3. a. 4. a. 6

_ 9

8

_ 12

b. b.

5. a. 2

_ 8

4

_ 10 6. a.

10

_ 12

5

_ 8

b. b.

7. a. 1

_ 4

5

_ 20 8. a.

5

_ 7

15

_ 21

b. b.

9. a. 10

_

16 4

_ 8 10. a. 3 _ 5 10 _ 15

b. b.

11. What pattern can you find in Parts a and b in the problems above?If the fractions are equivalent, the cross products are equal.

35

610

30 3010 ∗ 3 = = 5 ∗ 6

114 115

78

23

21 163 ∗ 7 = = 8 ∗ 2

23

69

18 189 ∗ 2 = = 3 ∗ 6

28

410

20 32 = 8 ∗ 410 ∗ 2 =

14

520

20 20 = 4 ∗ 520 ∗ 1=

48

1016

80 648 ∗ 10 = = 16 ∗ 4

69

812

72 7212 ∗ 6 = = 9 ∗ 8

58

1012

80 60 = 12 ∗ 58 ∗ 10 =

57

1521

105 10521 ∗ 5 = = 7 ∗ 15

35

1015

45 50 = 5 ∗ 1015 ∗ 3 =

Sample answer:

= 2

_ 3

6

_ 9

≠

=

≠

≠

=

≠

=

≠

Equivalent Fractions and Cross Products LESSON

8�3

278_323_EMCS_S_G6_U08_576442.indd 286 2/26/11 1:15 PM

Math Journal 2, p. 286

Student Page

Adjusting the Activity

704 Unit 8 Rates and Ratios

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 286)

Algebraic Thinking Go over the answers to Problems 1–10. Review the following:

Cross products are found by multiplying the numerator of each fraction by the denominator of the other fraction.

Cross multiplication is the process of finding cross products.

To support English language learners, demonstrate how an X is used to cross out a word or number. Relate this X to the terms cross products and cross multiplication.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

Discuss Problem 11. While there are several possible patterns, one stands out: If the fractions in Part a are equivalent, then the cross products in Part b are equal. If the fractions in Part a are not equivalent, then the cross products in Part b are not equal.

Point out that this pattern provides a way to test whether two fractions are equivalent. Have students use this rule to test several pairs of fractions for equivalence.

Suggestions:

3 _ 4 ? 9 _ 12 Cross products: 3 ∗ 12 = 36; 4 ∗ 9 = 36. The cross products are equal; therefore, the fractions are equivalent.

5 _ 6 ? 8 _ 9 Cross products: 5 ∗ 9 = 45; 6 ∗ 8 = 48. The cross products are not equal; therefore, the fractions are not equivalent.

3 _ 8 ? 1 _ 4 Not equivalent

16 _ 20 ? 12 _ 15 equivalent

Pose additional problems as needed.

ELL

3 _ 5 = 12 _ 20

5 _ 6 > 5 _ 8

7 _ 12 > 6 _ 11

6 _ 7 < 8 _ 9

14 _ 22 < 6 _ 8

10 _ 13 > 7 _ 11

Math MessageComplete the problems on journal page 286.

Study Link 8�2 Follow-UpBriefly go over answers. Have students share strategies for solving Problem 4.

Mental Math and Reflexes Students compare fractions using <, >, or =. Suggestions:

If time permits, have students share the strategies they used to compare the fractions.

Mathematical PracticesSMP1, SMP2, SMP4, SMP6, SMP7, SMP8Content Standards6.RP.3, 6.RP.3b, 6.RP.3d, 6.EE.5, 6.EE.7

704-709_EMCS_T_TLG2_G6_U08_L03_576922.indd 704 3/20/12 10:52 AM

Date Time

114 115

Solving Proportions with Cross ProductsLESSON

8� 3

Use cross multiplication to solve these proportions.

Example: 4 _ 6 = p _ 15

15 ∗ 4 = 6 ∗ p

60 = 6p

60 _ 6 = p

10 = p

1. 3

_ 6 =

y

_ 10

y = 5 2. 7

_ 21 =

3

_ c

c = 9

3. m

_ 20 =

2

_ 8

m = 5 4. 2

_ 10 =

5

_ z

z = 25

5. 9

_ 15 =

12

_ k k = 20 6.

10

_ 12 =

d

_ 9 d = 7.5

7. 2

_ 9 =

t

_ 54 t = 12 8.

4

_ 10 =

26

_ z z = 65

9. 3

_ 4 =

r

_ 28 r = 21 10.

16

_ p =

128

_ 40 p = 5

11. 51

_ 102 =

6

_ h h = 12 12.

j

_ 8 =

72

_ 192 j = 3

� �

�

�

�

�

46

p15

15 ∗ 4 = = 6 ∗ p

278_323_EMCS_S_G6_U08_576442.indd 288 2/21/11 4:41 PM


Student Page

Lesson 8�3 705

▶ Using Cross Products

WHOLE-CLASS ACTIVITY

to Solve ProportionsAlgebraic Thinking Write the following proportion on the board: 5 _ 6 = x _ 18 . Ask volunteers to explain how to solve the proportion using cross products. Students may suggest using the Identity Property of Multiplication, as was done in Lessons 8-1 and 8-2. This is correct; however, remind students that they are supposed to find a solution using cross products. If no one is able to do so, demonstrate the following approach:

Step 1 Cross multiply. Note that the cross product of 6 and x is written as 6 ∗ x, or 6x.

6 x or x 618 5 905––6

x––18

Step 2 Because we want the two fractions in the proportion to be equivalent, we also want the two cross products to be equal; that is, we want the product 6 ∗ x to equal the product 18 ∗ 5.

Step 3 Solve the equation from Step 2.

18 ∗ 5 = 6 ∗ x

90 = 6x

90 _ 6 = x

15 = x

Step 4 Write 15 in place of x in the proportion: 5 _ 6 = 15 _ 18 . Use cross multiplication to check that the two fractions are equivalent. 6 ∗ 15 = 90; 18 ∗ 5 = 90

Ongoing Assessment: Informing InstructionWatch for students who doubt the need to apply and practice the cross-products method because they can solve many of the problems in this lesson more quickly using other methods. Explain that the advantage of the cross-products method is that it works for all proportions, not just those with convenient numbers. To prove your point, pose a problem such as the following:

8.4 _ t = 11.2 _ 6.8 6.8 ∗ 8.4 = t ∗ 11.2

57.12 = 11.2t

5.1 = t

Adjusting the ActivityHave students use pencil and

paper or a calculator to calculate products as needed.



Solving Proportions with Cross Products continuedLESSON

8� 3

Date Time

114 115

For Problems 13–16, set up a proportion and solve it using cross multiplication. Show how the units cancel. Then write the answer.

Example: Jessie swam 6 lengths of the pool in 4 minutes. At this rate, how many lengths will she swim in 10 minutes?

Proportion:

10 minutes ∗ 6 lengths = 4 minutes ∗ n lengths

60 minutes ∗ lengths = 4 minutes ∗ n lengths

= n lengths

15 lengths = n lengths

Answer: Jessie will swim 15 lengths in 10 minutes.

13. Belle bought 8 yards of ribbon for $6. Solution:How many yards could she buy for $9?

Answer: Belle could buy yards of ribbon for $9.

$9 ∗ 8 yards = $6 ∗ n yards $72 ∗ yards = $6 ∗ n yards

$72 ∗ yards _ $6 = n yards

12 yards = n yards

=

8 yards n yards$6 $9

12

6 lengths

4 minutes

n lengths

10 minutes=

Solution: 6 _ 4 = n _ 10

64

n10

10 ∗ 6 = = 4 ∗ n

60 minutes ∗ lengths4 minutes

278_323_EMCS_S_G6_MJ2_U08_576442.indd 289 3/9/11 11:11 AM


Student Page

Date Time

LESSON

8�3

14. Before going to France, Maurice Solution: exchanged $25 for 20 euros. At that exchange rate, how many euroscould he get for $80?

Answer: Maurice could get euros for $80.

15. One gloomy day, 4 inches of rain Solution:fell in 6 hours. At this rate, how many inches of rain had fallen after 4 hours?

Answer: inches of rain had fallen in 4 hours.

16. Adelio’s apartment building has Solution: 9 flights of stairs. To climb to the top floor, he must go up 144 steps. How many steps must he go up to climb 5 flights?

Answer: Adelio must climb steps.

=

=

9 flights 5 flights144 steps s steps

=

4 inches p inches6 hours 4 hours

64

2. _ 6

80

$25 $80x euros20 euros


8�3

x euros ∗ $25 = 20 euros ∗ $80x euros ∗ $25 = $1,600 ∗ euros x euros = $1,600 ∗ euros

_ $25 x euros = 64 euros

4 hours ∗ 4 inches = 6 hours ∗ p inches 16 hours ∗ inches = 6 hours ∗ p inches

16 hours ∗ inches __ 6 hours = p inches 2.

_ 6 inches = p inches

s steps ∗ 9 flights = 144 steps ∗ 5 flightss steps ∗ 9 flights = 720 steps ∗ flights

s steps = 720 steps ∗ flights __ 9 flights

s steps = 80 steps

289A_289B_EMCS_S_G6_MJ2_U08_576442.indd 289A 3/9/11 11:13 AM

Math Journal 2, p. 289A

Student Page

Adjusting the Activity


Guide students in solving a few more proportions using Steps 1–4.

Example: 6 _ 48 = 8 _ n

Cross multiply: n ∗ 6 = 48 ∗ 8

Solve: 6n = 384

n = 384 _ 6

n = 64

Replace thevariable: 6 _ 48 = 8 _ 64

Check: 64 ∗ 6 = 48 ∗ 8

Suggestions:

6 _ 9 = x _ 12 15 _ 20 = 9 _ r z _ 3 = 1 _ 5

x = 8 r = 12 z = 0.6

Use a quick common denominator (QCD) or multiplicative inverses to explain why cross multiplication works:

3 _ 8 = 12 _ 32

Find the QCD: Multiply both sides by 8 ∗ 32.

32 º 8 º 38

8 º 32 º 1232

32 º 3 8 º 12 cross products

Multiplicative inverses: Rewrite the proportion as 3 ∗ 1 _ 8 = 12 ∗ 1 _ 32 and multiply both sides by the multiplicative inverses of 1 _ 8 and 1 _ 32 .

(3 ∗ 1 _ 8 ) ∗ 8 ∗ 32 = (12 ∗ 1 _ 32 ) ∗ 8 ∗ 32

3 ∗ 32 = 12 ∗ 8 cross products


▶ Solving Problems Using PARTNER ACTIVITY

Cross Multiplication(Math Journal 2, pp. 288, 289, 289A, and 289B)

Algebraic Thinking Assign journal page 288. When most students have completed the problems, bring the class together and go over the answers.

Ongoing Assessment: JournalPage 288 � Problems 1– 6Recognizing Student Achievement

Use journal page 288, Problems 1–6 to assess students’ ability to use cross products to write an open number sentence. Students are making adequate progress if they are able to write open number sentences for Problems 1–6. Some students may be able to solve mentally for missing variables. [Operations and Computation Goal 6]

PROBLEMBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEMMMBLELBLELBLLLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMMOOOOOOOOOOOBBBBBBLBLBLBLBLLLLLPROPROPROPROPROPROPROPROPROPRPROPPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROOROOPPPPPPP MMMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEEEEEELELEELEEEEEEEEELLLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBB EELEMMMMMMMOOOOOOOOOBBBLBLBLBLBBLBBOOOROROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLVINVINVINVINVINNNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOLOLOLOO VINVINVVINLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGOLOLOLOLOLOLOLOLOOO VVVVLLLLLLLLLLVVVVVVVVVVOOSOSOOSOSOSOSOSOSOSOOSOSOSOSOSOOOOOSOSOSOSOSOSSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIISOLVING

704-709_EMCS_T_TLG2_G6_U08_L03_576922.indd 706 3/10/11 3:42 PM

Date Time


8�3Set up a proportion for each problem and solve it using cross multiplication.

17. Sarah uses 5 scoops of coffee beans to brew Solution:8 cups of coffee. How many scoops of beans does Sarah use per cup?

Answer: Sarah uses scoop(s) of beans per cup of coffee.

18. Jeremiah ran 1 1 _ 4 miles in 12 minutes. At this Solution:pace, how long would it take him to run 5 miles?

Answer: It would take Jeremiah minutes to run 5 miles.

19. It took Zach 12 days to read a book that was Solution:186 pages long. If he read the same amount each day, how many pages did he read in one week?

Answer: Zach read pages in one week.

20. At sea level, sound travels 0.62 mile in 3 seconds. Solution: What is the speed of sound in miles per hour? (Hint: First find the number of seconds in 1 hour.)

Answer: Sound travels at the rate of miles per hour.

3,600 ∗ 0.62 = 3d 2,232 = 3d 2,232

_ 3 = 3

_ 3 d

744 = d

=

0.62 mile d miles3 sec 3,600 sec

744

1 ∗ 5 = 8s 5 = 8s 5 _ 8 = s

=

5 scoops s scoops8 cups 1 cup

5 _ 8

48

108.5

1 1 _ 4 m = 12 ∗ 51 1 _ 4 m = 60

m = 60 _ 1 1 _ 4

m = 48

m minutes=

1 1 _

4 miles 5 miles12 minutes

12p = 186 ∗ 7 12p = 1,302 p = 1,302

_ 12 p = 108.5

p pages=

12 days 7 days186 pages

114 115

289A_289B_EMCS_S_G6_MJ2_U08_576442.indd 289B 3/9/11 11:13 AM

Math Journal 2, p. 289B

Student Page

Math Boxes

2. A boat traveled 128 kilometers in 4 hours.

Fill in the rate table.

At this rate, how far did the boat travel in 2 hours 15 minutes?

72 km

LESSON

8�3

Date Time

5. Insert parentheses to make each number sentence true.

a. 0.01 ∗ 7 + 9 / 4 = 0.04

b. 4 _ 5 ∗ 25 - 10 / 2 = 15

c. √ _

64 / 5 + 3 ∗ 3 = 3

d. 5 ∗ 102 + 102 ∗ 2 = 700

1. Which rate is equivalent to 70 km in 2 hr 30 min? Fill in the circle next to the best answer.

A. 35 km in 75 min

B. 70,000 m in 230 min

C. 140 km in 4 hr 30 min

D. 1,400 m in 300 min

3. A bag contains 1 red counter, 2 blue counters, and 1 white counter. You pick 1 counter at random. Then you pick a second counter without replacing the first counter.

a. Draw a tree diagram to show all possible counter combinations.

b. What is the probability of picking 1 red counter and 1 white counter (in either order)?

4. Add or subtract.

a. -303 + (-28) =

b. = 245 - 518

c. = -73 + 89

d. 280 - (-31) =

110 111

156

109–111

95 96 247

( )

( )

( )

( ) ( )

R

W B1 B2 R B1 B2 R W B2 R W B1

B1W B2

-331-273

16311

2 _ 12 , or 1 _ 6

distance (km) 24 72 144

hours 3 _ 4 1 1 _

2 32 1

_ 4 48 96

4 1 _ 2

278_323_EMCS_S_G6_U08_576442.indd 287 2/26/11 1:15 PM


Student Page

Links to the Future

Lesson 8�3 707

Work through the example at the top of journal page 289 with the class. Show students how the units in the problem function much like numbers when they are included in the computation. Just as a number divided by itself is equal to 1, a unit divided by itself is also equal to 1. It is sometimes said that the units “cancel” and they can simply be crossed out as shown below.

60 minutes ∗ lengths __ 4 minutes = n lengths

NOTE This is a simple example of a strategy called dimensional analysis. Students will use dimensional analysis in future mathematics and science courses. It is not necessary to introduce the term at this time.

Have students complete journal pages 289 and 289A, showing how the units cancel.

After most students have finished these problems, ask them why keeping track of the units is a useful strategy. Sample answers: It helps ensure that I have set up the proportion correctly. It helps me see the correct unit to use in my answer. Tell students that it is not necessary for them to include units every time they solve a proportion with cross multiplication, but it is a good strategy to use to check their work or help them on more difficult problems.

Have students solve the problems on journal page 289B. It is not necessary for them to include units in their work on these problems, but they may do so if they wish.

Students will apply their knowledge of cross products in future algebra and science courses. It is important that they be able to use cross products to write open number sentences.

2 Ongoing Learning & Practice

▶ Playing Fraction/Whole Number PARTNER ACTIVITY

Top-It(Student Reference Book, pp. 319 and 320)

Distribute four each of number cards 1–10 (from the Everything Math Deck, if available) to each partnership.

Students use cards to form whole numbers and fractions. They then find and compare the products.

▶ Math Boxes 8�3

INDEPENDENT ACTIVITY

(Math Journal 2, p. 287)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-1. The skills in Problems 4 and 5 preview Unit 9 content.


Teaching MasterName Date Time

LESSON

8 � 3 Double Number Lines

Howie is making tamales. He used 8 cups of filling to make 4 dozen tamales. How much filling does he need to make 10 dozen tamales?

The double number line below can be used to help solve this problem. Notice that the scale at the top of the number line is labeled in dozens of tamales. The scale at the bottom of the number line is labeled in cups of filling. Find the mark for 8 cups of filling. Notice how it lines up with 4 dozen tamales. This represents the information given in the problem.

The per-unit rate is also shown on the number line: Howie uses 2 cups of filling per 1 dozen tamales, so the mark for 1 dozen tamales lines up with the mark for 2 cups of filling. This information was used to complete the double number line.

Dozens of tamales

Cups of filling

0

0

1

2

2

4

3

6

4

8

5

10

6

12

7

14

8

16

9

18

10

20

The problem asks how much filling Howie needs to make 10 dozen tamales. Find the mark for 10 dozen tamales. Then find the number on the cups-of-filling scale that lines up with this mark. It’s 20, so Howie needs 20 cups of filling to make 10 dozen tamales.

Use double number lines to help you solve the problems.

1. A marine animal trainer noted that the aquarium’s newest beluga whale ate 150 pounds of food in 3 days. The whale was fed the same amount of food each day.

a. How many pounds of food does the whale eat per day? 50 pounds

b. Use your answer to Part a to fill in the blanks on the top scale of the double number line below.

35025020010050Pounds of food

Days

0

0 1 2 3 4 5 6 7

300150

c. If he continues to eat at this rate, how many pounds of food will the whale eat in 5 days? 250 pounds

249A-249B_EMCS_B_MM_G6_U08_576981.indd 249A 3/9/11 12:10 PM

Math Masters, p. 249A

pyg

gp

STUDY LINK

8�3 Calculating Rates

Name Date Time

If necessary, draw a picture, find a per-unit rate, make a rate table, or use a proportion to help you solve these problems.

1. A can of worms for fishing costs $2.60. There are 20 worms in a can.

a. What is the cost per worm?

b. At this rate, how much would 26 worms cost?

2. An 11-ounce bag of chips costs $1.99.

a. What is the cost per ounce, rounded to the nearest cent?

6. A 1-pound bag of candy containing 502 pieces costs 16.8 cents per ounce. What is the cost of 1 piece of candy? Circle the best answer.

1.86 cents 2.99 cents 0.33 cent cent

7. Mr. Rainier’s car uses about 1.6 fluid ounces of gas per minute when the engine is idling. One night, he parked his car but forgot to turn off the motor. He had just filled his tank. His tank holds 12 gallons.

About how many hours will it take before his car runs out of gas?

Explain what you did to find the answer.

Sources: 2201 Fascinating Facts; Everything Has Its Price

b. What is the cost per pound, rounded to the nearest cent?

3. Just 1 gram of venom from a king cobra snake can kill 150 people. At this rate, about how many people would 1 kilogram kill?

4. A milking cow can produce nearly 6,000 quarts of milk each year. At this rate, about how many gallons of milk could a cow produce in 5 months?

5. A dog-walking service costs $2,520 for 6 months.

What is the cost for 2 months? For 3 years?

111–116

Try This

150,000 people

$2.88$0.18 per oz

$3.38

625 gallons

$840 $15,120

1 _ 2

Sample answer: 128 oz = 1 gal; 12 gal = 1,536 oz;

16 hours

$0.13 per worm

1,536 oz _ 1.6 oz per min = 960 min; 960 min

_ 60 min per hour = 16 hours

246-284_EMCS_B_G6_MM_U08_576981.indd 249 2/28/11 1:02 PM

Math Masters, p. 249

Study Link Master


▶ Study Link 8�3


(Math Masters, pp. 249 and 251)

Home Connection Students solve rate problems on Math Masters, page 249.

If you haven’t already done so, review the instructions for Math Masters, page 251 with the class. Students may postpone completing Parts B and C of the table until after they have completed Lesson 8-4. If a grocery store posts a unit price, ask students to check that the price is accurate.

3 Differentiation Options

READINESS

SMALL-GROUP ACTIVITY

▶ Solving Equations (ax = b) 5–15 Min

To provide experience solving equations of the form ax = b, have students review and practice solving equations using the method of their choice.

Suggestions:

6 ∗ g = 54 g = 9 9m = 15 ∗ 12 m = 20

10 ∗ y = 35 y = 3.5 35k = 70(125) k = 250

180 = 15 ∗ t t = 12 90 = 12j j = 7.5

5x = 80 ∗ 10 x = 160 3f = 0.62(300) f = 62

29(3) = 3p p = 29 1 _ 5 w = 20 1 _ 2 w = 102.5

ENRICHMENT PARTNER ACTIVITY

▶ Using Double Number Lines 15–30 Min

(Math Masters, pp. 249A and 249B)

Students explore an alternative way to solve rate problems by using double number lines. Tell students that a double number line is a number line that has two scales: one above the line and one below the line. Have students look at the double number line at the top of Math Masters, page 249A. Point out the two scales: dozens of tamales above, and cups of filling below.

Have students read the top of Math Masters, page 249A with a partner. Ask them to locate the per-unit rate on the number line and discuss how it can help to determine the scales on each side of the number line. Then have partnerships solve the problems on Math Masters, pages 249A and 249B.

704-709_EMCS_T_TLG2_G6_U08_L03_576922.indd 708 3/10/11 3:42 PM

Teaching Master

LESSON

8�3

Name Date Time

Ingredients for Peanut Butter Fudge

1. The list at the right shows the ingredients used to make peanut butter fudge but not how muchof each ingredient is needed. Use the following clues to calculate the amount of each ingredient needed to make 1 pound of peanut butter fudge. Record eachamount in the ingredient list.

Clues

� Use 20 cups of sugar to make 10 pounds of fudge.

� You need cups of milk to make 5 pounds of fudge.

� You need 15 cups of peanut butter to make 48 poundsof fudge. (Hint: 1 cup = 16 tablespoons)

� An 8-pound batch of fudge uses 1 cup of corn syrup.

� Use 6 teaspoons of vanilla for each 4 pounds of fudge.

� Use teaspoon of salt for each 4 pounds of fudge.

2. Suppose you wanted to make an 80-pound batch of fudge. Record how much of each ingredient you would need.

Use the following equivalencies and your ingredient lists to complete each problem.

� 3 teaspoons = 1 tablespoon

� 16 tablespoons = 1 cup

3. cups of peanut butter are needed for 80 pounds of fudge.

4. 10 cups of corn syrup are needed for 80 pounds of fudge.

5. tablespoons of vanilla are needed for 80 pounds of fudge.40

25

Ingredient List for 80 Pounds of Peanut Butter Fudge

cups of sugar tablespoons of corn syrup

cups of milk teaspoons of vanilla

tablespoons of peanut butter teaspoons of salt

60400

12010

3 3 _ 4

1 _ 2

Peanut Butter Fudge(makes 1 pound)

cups of sugar

cup of milk

tablespoons ofpeanut butter tablespoons ofcorn syrup teaspoons of vanilla

teaspoon of salt

2

52

3 _ 4

1 1 _ 2 1 _ 8

160 160

246-284_EMCS_B_G6_MM_U08_576981.indd 250 2/28/11 1:02 PM

Math Masters, p. 250

Name Date Time

Double Number Lines continuedLESSON

8 � 3

For Problems 2–4, fill in the blanks on the double number lines and use them to help you solve the problem.

2. Jamie is ordering supplies for his dog-washing business. Last week, he washed 24 dogs and used 4 bottles of shampoo. Jamie uses the same amount of shampoo for each dog he washes.

427

366

305

24183

122

61

Dogs

Bottles of shampoo

0

0 4

a. How many dogs can he wash with one bottle of shampoo? 6 dogs

b. How many bottles of shampoo should he order if he expects to wash 30 dogs this week? 5 bottles

3. A craft store has skeins of yarn on special. They are selling 2 skeins for $5.

7654321$17.50$12.50$2.50 $7.50 $10 $15

Skeins

Cost

0

$0 $5

a. What is the cost per skein of yarn? $2.50 b. Holly needs 6 skeins of yarn to make an afghan. How much will the

yarn cost? $154. Katie rode her bicycle to work today. The 8-mile ride took her 40 minutes.

1155

1050

94540

735

630

525

420

315

210

15

Miles

Minutes

0 8

0

a. On average, how long does it take Katie to ride one mile? 5 minutes

b. At that rate, how long will it take her to ride 11 miles to get from work to her sister’s house? 55 minutes

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Math Masters, p. 249B

Teaching Master

Lesson 8�3 709

EXTRA PRACTICE


▶ Calculating Ingredient Amounts 15–30 Min

(Math Masters, p. 250)

Students practice solving rate problems by calculating how much of each ingredient is needed to make 1 pound and 80 pounds of peanut butter fudge.

ELL SUPPORT

SMALL-GROUP ACTIVITY

▶ Illustrating Terms 15–30 Min

To provide language support for solving proportions, have students create a poster that features the steps for using cross products to solve proportions. Their poster should include the terms cross products and cross multiplication.

Planning Ahead

Remind students to collect nutrition labels from containers of food, such as cans of soup, cups of yogurt, and cereal boxes. They will need to bring these labels to school for use in Lesson 8-5.

If you haven’t already done so, provide students with a copy of Study Link 8-4 (Math Masters, page 251) and remind them to collect data about the cost and weight of the listed items. They may postpone the calculations of unit price until after they have completed Lesson 8-4.


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LESSON

8 � 3 Double Number Lines

Howie is making tamales. He used 8 cups of filling to make 4 dozen tamales. How much filling does he need to make 10 dozen tamales?

The double number line below can be used to help solve this problem. Notice that the scale at the top of the number line is labeled in dozens of tamales. The scale at the bottom of the number line is labeled in cups of filling. Find the mark for 8 cups of filling. Notice how it lines up with 4 dozen tamales. This represents the information given in the problem.

The per-unit rate is also shown on the number line: Howie uses 2 cups of filling per 1 dozen tamales, so the mark for 1 dozen tamales lines up with the mark for 2 cups of filling. This information was used to complete the double number line.

Dozens of tamales

Cups of filling

0

0

1

2

2

4

3

6

4

8

5

10

6

12

7

14

8

16

9

18

10

20

The problem asks how much filling Howie needs to make 10 dozen tamales. Find the mark for 10 dozen tamales. Then find the number on the cups-of-filling scale that lines up with this mark. It’s 20, so Howie needs 20 cups of filling to make 10 dozen tamales.

Use double number lines to help you solve the problems.

1. A marine animal trainer noted that the aquarium’s newest beluga whale ate 150 pounds of food in 3 days. The whale was fed the same amount of food each day.

a. How many pounds of food does the whale eat per day? pounds

b. Use your answer to Part a to fill in the blanks on the top scale of the double number line below.

Pounds of food

Days

0

0 1 2 3 4 5 6 7

300150

c. If he continues to eat at this rate, how many pounds of food will the whale eat in 5 days? pounds

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Double Number Lines continuedLESSON

8 � 3

For Problems 2–4, fill in the blanks on the double number lines and use them to help you solve the problem.

2. Jamie is ordering supplies for his dog-washing business. Last week, he washed 24 dogs and used 4 bottles of shampoo. Jamie uses the same amount of shampoo for each dog he washes.

Dogs

Bottles of shampoo

0

0 4

a. How many dogs can he wash with one bottle of shampoo? dogs

b. How many bottles of shampoo should he order if he expects to wash 30 dogs this week? bottles

3. A craft store has skeins of yarn on special. They are selling 2 skeins for $5.

Skeins

Cost

0

$0 $5

a. What is the cost per skein of yarn?

b. Holly needs 6 skeins of yarn to make an afghan. How much will the yarn cost?

4. Katie rode her bicycle to work today. The 8-mile ride took her 40 minutes.

Miles

Minutes

0 8

0

a. On average, how long does it take Katie to ride one mile? minutes

b. At that rate, how long will it take her to ride 11 miles to get from work to her sister’s house? minutes

249A-249B_EMCS_B_MM_G6_U08_576981.indd 249B 4/1/11 11:45 AM

Solving Proportions by Cross Multiplication · PDF file[Operations and Computation Goal 2] ... per partnership: ... Algebraic Thinking Go over the answers to Problems 1–10. Review

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